**Student Mathematical Library**

Volume: 84;
2017;
312 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-1-4704-3653-7
Product Code: STML/84**

List Price: $52.00

Individual Price: $41.60

**Electronic ISBN: 978-1-4704-4125-8
Product Code: STML/84.E**

List Price: $52.00

Individual Price: $41.60

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#### Supplemental Materials

# A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond \(\mathbb{Z}\)

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*Paul Pollack*

Gauss famously referred to mathematics as the “queen of the
sciences” and to number theory as the “queen of
mathematics”. This book is an introduction to algebraic
number theory, meaning the study of arithmetic in finite extensions of
the rational number field \(\mathbb{Q}\). Originating in the
work of Gauss, the foundations of modern algebraic number theory are
due to Dirichlet, Dedekind, Kronecker, Kummer, and others. This book
lays out basic results, including the three “fundamental
theorems”: unique factorization of ideals, finiteness of the
class number, and Dirichlet's unit theorem. While these theorems are
by now quite classical, both the text and the exercises allude
frequently to more recent developments.

In addition to traversing the main highways, the book reveals some
remarkable vistas by exploring scenic side roads. Several topics
appear that are not present in the usual introductory texts. One
example is the inclusion of an extensive discussion of the theory of
elasticity, which provides a precise way of measuring the failure of
unique factorization.

The book is based on the author's notes from a course delivered at
the University of Georgia; pains have been taken to preserve the
conversational style of the original lectures.

#### Readership

Undergraduate and graduate students interested in algebraic number theory.

#### Reviews & Endorsements

This is a lucid, clearly written text, with a thoughtful choice and arrangement of topics, presented with contagious enthusiasm. It is a welcome addition to the existing literature on the subject.

-- Charles Helou, Mathematical Reviews

#### Table of Contents

# Table of Contents

## A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond $\mathbb{Z}$

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Chapter 1. Getting our feet wet 112
- Chapter 2. Cast of characters 920
- Chapter 3. Quadratic number fields: First steps 1930
- Chapter 4. Paradise lost —and found 2738
- Chapter 5. Euclidean quadratic fields 3748
- Chapter 6. Ideal theory for quadratic fields 4960
- Chapter 7. Prime ideals in quadratic number rings 5970
- Chapter 8. Units in quadratic number rings 6778
- Chapter 9. A touch of class 7990
- Chapter 10. Measuring the failure of unique factorization 91102
- Chapter 11. Euler’s prime-producing polynomial and the criterion of Frobenius–Rabinowitsch 105116
- Chapter 12. Interlude: Lattice points 117128
- Chapter 13. Back to basics: Starting over with arbitrary number fields 129140
- Chapter 14. Integral bases: From theory to practice, and back 143154
- Chapter 15. Ideal theory in general number rings 159170
- Chapter 16. Finiteness of the class group and the arithmetic of \Z 171182
- Chapter 17. Prime decomposition in general number rings 179190
- Chapter 18. Dirichlet’s unit theorem, I 195206
- Chapter 19. A case study: Units in \Z[√[3]2] and the Diophantine equation 𝑋³-2𝑌³=±1 205216
- Chapter 20. Dirichlet’s unit theorem, II 215226
- Chapter 21. More Minkowski magic, with a cameo appearance by Hermite 225236
- Chapter 22. Dedekind’s discriminant theorem 241252
- Chapter 23. The quadratic Gauss sum 255266
- Chapter 24. Ideal density in quadratic number fields 271282
- Chapter 25. Dirichlet’s class number formula 281292
- Chapter 26. Three miraculous appearances of quadratic class numbers 295306
- Index 313324
- Back Cover Back Cover1329